I have a WLAN and can attach a hard disk directly to the router (Fritzbox) via an USB interface.
I have a Noxon 2 Radio for iPod. The fact that I can listen to music from the web without running the computer is great! I hate the vent sound...

I know that I can attach the "music"-harddisk directly to the Noxon via USB, but I'd love to have the search abilities which Twonky gives me, which is not the case when the Computer is not running in parallel.

Therefore:

Is there a way to run "my" Twonky by installing it on a server of a internet provider and make it contact my harddisk locally and then sent the stream to my Noxon via WLAN. Under no circumstances must my harddisk be readable to outsiders as this might lead to copyright claims.

I've never heard about such an internet service you are describing. But maybe you want to give the mp3tunes service from my-noxon.net a try. You can listen to your music directly from the internet without the need of having a PC or a NAS running. Search abilities are not so extensive as with Twonky though.

More to try:

If your Fritzbox is a 7170 you can install the latest "labor firmware" from AVM which includes a music server such as Twonky. And if you want to have "the original" you can even install Twonky on your 7170:

I myself just own a Fritzbox 3070 and connected it to a NSLU2 NAS with Twonky on it. The NSLU is absolutely quiet (no vent), needs little power and I keep it running almost all the time together with the Fritzbox. Perfect solution for me.